State variable states

Incorrect readings of appropriate state variables State variables which are considered appropriate are constant State variables which are considered appropriate may change within the batch and batch-to-batch... 

Energy-per-entity and number of entities enter in the category of state variables. State variables are defined as those variables that it is necessary to know the value and their mutual relations to help determine the amount of energy and therefore the state of the system. They form a pair as each subvariety possesses two of these state variables, also called conjugate state variables. 

In classical mechanics, the state of the system may be completely specified by the set of Cartesian particle coordinates r. and velocities dr./dt at any given time. These evolve according to Newton s equations of motion. In principle, one can write down equations involving the state variables and forces acting on the particles which can be solved to give the location and velocity of each particle at any later (or earlier) time t, provided one knows the precise state of the classical system at time t. In quantum mechanics, the state of the system at time t is instead described by a well behaved mathematical fiinction of the particle coordinates q- rather than a simple list of positions and velocities. 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

Relationships from thennodynamics provide other views of pressure as a macroscopic state variable. Pressure, temperature, volume and/or composition often are the controllable independent variables used to constrain equilibrium states of chemical or physical systems. For fluids that do not support shears, the pressure, P, at any point in the system is the same in all directions and, when gravity or other accelerations can be neglected, is constant tliroughout the system. That is, the equilibrium state of the system is subject to a hydrostatic pressure. The fiindamental differential equations of thennodynamics ... 

The transition elements are often said to exhibit variable valency. Because they so readily form complex compounds, it is better to use the term variety of oxidation states . The states usually found for the elements Sc-Zn are ... 

We consider a two state system, state A and state B. A state is defined as a domain in phase space that is (at least) in local equilibrium since thermodynamic variables are assigned to it. We assume that A or B are described by a local canonical ensemble. There are no dark or hidden states and the probability of the system to be in either A or in B is one. A phenomenological rate equation that describes the transitions between A and B is... 

To extend the above kinetie model to this more general ease in whieh degenerate levels oeeur, one uses the number of moleeules in eaeh level (N and Nf for the two levels in the above example) as the time dependent variables. The kinetie equations then governing their time evolution ean be obtained by summing the state-to-state equations over all states in eaeh level... 

Since G is a state variable and forms exact differentials, an alternative expression for dG is... 

While this is an easy calculation to make, Eq. (3.7) does little to clarify exactly what AS means. Phenomenological proofs that AS as defined by Eq. (3.7) is a state variable often leave us with little more than a lament for the inefficiency of spontaneous processes. 

Reaction (5. EE) is particularly useful for the discussion of thermodynamic considerations because of the way differences in thermodynamic state variables are independent of path. Accordingly, if we know the value of AG for reaction (5. EE), we have characterized the following ... 

State Functions State functions depend only on the state of the system, not on past history or how one got there. If r is a function of two variables, x and y, then z x,y) is a state function, since z is known once X and y are specified. The differential of z is... 

In process simulation it is necessary to calculate enthalpy as a function of state variables. This is done using the following formulas, derived from the above relations by considering S and H as functions of T and p. 

Fiypothesis unable to explain the values of the state variables observed... 

K set of referential internal state variables I velocity gradient tensor... 

The theory is initially presented in the context of small deformations in Section 5.2. A set of internal state variables are introduced as primitive quantities, collectively represented by the symbol k. Qualitative concepts of inelastic deformation are rendered into precise mathematical statements regarding an elastic range bounded by an elastic limit surface, a stress-strain relation, and an evolution equation for the internal state variables. While these qualitative ideas lead in a natural way to the formulation of an elastic limit surface in strain space, an elastic limit surface in stress space arises as a consequence. An assumption that the external work done in small closed cycles of deformation should be nonnegative leads to the existence of an elastic potential and a normality condition. 

Specific applications of the theory are not considered in this chapter. Only one example, that of small deformation classical plasticity, is worked out in Section 5.3. The set of internal state variables k is taken to be comprised of... 

As with any constitutive theory, the particular forms of the constitutive functions must be constructed, and their parameters (material properties) must be evaluated for the particular materials whose response is to be predicted. In principle, they are to be evaluated from experimental data. Even when experimental data are available, it is often difficult to determine the functional forms of the constitutive functions, because data may be sparse or unavailable in important portions of the parameter space of interest. Micromechanical models of material deformation may be helpful in suggesting functional forms. Internal state variables are particularly useful in this regard, since they may often be connected directly to averages of micromechanical quantities. Often, forms of the constitutive functions are chosen for their mathematical or computational simplicity. When deformations are large, extrapolation of functions borrowed from small deformation theories can produce surprising and sometimes unfortunate results, due to the strong nonlinearities inherent in the kinematics of large deformations. The construction of adequate constitutive functions and their evaluation for particular... 

Motivated by the qualitative observations made above, a set of internal state variables deseribing the internal strueture of the material will be intro-dueed ab initio, denoted eolleetively by k. Their physieal meaning or preeise properties need not be established at this point, and they may inelude sealar, veetor, or tensor quantities. The following eonstitutive assumptions are now made ... 

Inelastic Loading. The strain lies on the elastic limit surface = 0, and the tangent to the strain history points in a direction outward from the elastic limit surface > 0. The material is said to be undergoing inelastic loading, and k is assumed to be a function of the strain s, the internal state variables k, and the strain rate k... 

This implies that, at constant k, the line integral of the differential form s de, parametrized by time t, taken over the closed curve h) zero. This is the integrability condition for the existence of a scalar function tj/ e) such that s = d j//de (see, e.g., Courant and John [13], Vol. 2, 1.10). This holds for an elastic closed cycle at any constant values of the internal state variables k. Therefore, in general, there exists a function ij/... 

The normality conditions (5.56) and (5.57) have essentially the same forms as those derived by Casey and Naghdi [1], [2], [3], but the interpretation is very different. In the present theory, it is clear that the inelastic strain rate e is always normal to the elastic limit surface in stress space. When applied to plasticity, e is the plastic strain rate, which may now be denoted e", and this is always normal to the elastic limit surface, which may now be called the yield surface. Naghdi et al. by contrast, took the internal state variables k to be comprised of the plastic strain e and a scalar hardening parameter k. In their theory, consequently, the plastic strain rate e , being contained in k in (5.57), is not itself normal to the yield surface. This confusion produces quite different results. 

The remainder of this section will be concerned with a particular case in which normality conditions hold. The constitutive equation for the internal state variables (5.11) involves the constitutive function a, and the normality conditions (5.56) and (5.57) involve an unknown scalar factor y. In some circumstances, a may be eliminated and y may be evaluated by using the consistency condition. These circumstances arise if b is nonsingular so that the normality condition in strain space (5.56j) may be solved for k... 

In this section, the general inelastic theory of Section 5.2 will be specialized to a simple phenomenological theory of plasticity. The inelastic strain rate tensor e may be identified with the plastic strain rate tensor e . In order to include isotropic and kinematic hardening, the set of internal state variables, denoted collectively by k in the previous theory, is reduced to the set (k, a) where k is a scalar representing isotropic hardening and a is a symmetric second-order tensor representing kinematic hardening. The elastic limit condition in stress space (5.25), now called a yield condition, becomes... 

Prager s rule of kinematic hardening is expressed by a = ce where c is a constant. Generalizing these concepts, the evolution equations for the internal state variables will be taken in the form... 

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